Replica method

The crux of the replica trick is that while the disorder averaging is done assuming ${\displaystyle n}$ to be an integer, to recover the disorder-averaged logarithm one must send ${\displaystyle n}$ continuously to zero. This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the same results. (A natural sufficient rigorous proof that the replica trick works would be to check that the assumptions of Carlson's theorem hold, especially that the ratio ${\displaystyle (Z^{n}-1)/n}$ is of exponential type less than π.)